Composite parts for various industrial applications are often manufactured by “laying up” composite material onto a piece of tooling (also referred to herein as a “tool”) that serves as a mold. The material is cured (e.g., hardened) in place on the tooling, and the resulting shaped part is removed from the tooling after the material is cured. Different shaped parts present different manufacturing challenges. Existing methods and tooling for manufacturing parts, such as ducts with one or more integral flanges can be technically and/or ergonomically challenging to perform. For example, existing tooling for forming such flanged ducts consists of a cylindrical form with two planar plates attached thereto (one at either end of the cylindrical form) and arranged at an angle to one another. The intersections of the plates and the cylindrical form create a “female” fillet radius at each end of the cylindrical form, and composite material is laid up around the outside of the cylindrical form and against the female fillet radius and onto the plates, in order to form the flanges. Such tooling requires cutting and shaping many pieces of composite material of various shapes, fitting the pieces together on the tooling, and forming the material inside a female fillet radius where there is very little working space (e.g., the part is formed on an outer surface of the cylindrical form of the tooling and material must be laid inside the fillet radius formed between the cylindrical form and the plates). This can lead to difficult, expensive, and time-consuming techniques with unpredictable results and/or flawed parts having bridging, wrinkling, and/or high rejection rates.